Properties of unit impulse signal
- The unit impulse signal is an even signal . The mathematical expression to determine the unit impulse signal is even signal is shown as :
[Tex]\delta(t)=\delta(-t)[/Tex] in continuous time domain
[Tex]\delta[n]=\delta[-n][/Tex] in discrete time domain
- Area under the unit impulse signal is equal to one , this property represent the energy or amplitude of the unit impulse signal to normalized to 1 .
[Tex]\int_{-\infty }^{\infty } \delta (t) dt = 1[/Tex] in continuous time domain
[Tex]\sum_{n=-\infty }^{\infty } \delta [n] = 1 [/Tex]in discrete time domain
- The scaling property can be represented in many ways . Scaling property is the property that will compress or decompress the signal by a constant . Let us look at the each way representation of scaling property :
1. [Tex]\int_{-\infty }^{\infty } C . \delta (t) dt = C [/Tex] in continuous time domain where C is a constant .
[Tex]\sum_{n=-\infty }^{\infty } C . \delta [n] = C[/Tex] in discrete time domain where C is a constant .
2. [Tex]\int_{-\infty }^{\infty } \delta (Ct) dt = \frac{1}{C} \delta (t)[/Tex] in continuous time domain where C is a constant .
[Tex] \sum_{n=-\infty }^{\infty } \delta [C.n] = \frac{1}{C} \delta[n][/Tex] in discrete time domain where C is a constant
3. [Tex]\int_{-\infty }^{\infty } \delta (\frac{t}{C}) dt = C .\delta (t)[/Tex] in continuous time domain where C is a constant .
[Tex]\sum_{n=-\infty }^{\infty } \delta [n/C] = C. \delta[n] [/Tex] in discrete time domain where C is a constant .
- Shifting property is the property of the signal that will shift the signal either right or left of the origin by a constant unit . Let us look at the mathematical representation of the shifting property in continuous time domain and discrete time domain.
[Tex]\int_{-\infty }^{\infty } \delta (t-to) dt = \delta (to)[/Tex] in continuous time domain where to is the constant that the signal shifted.
The value of to will be either positive or negative based on the value of to the signal shifted to left or right of the origin.
[Tex] \sum_{n=-\infty }^{\infty } \delta [n-K] = \delta[K][/Tex] in discrete time domain where K is the constant that the signal shifted.
The value of to will be either positive or negative based on the value of to the signal shifted to left or right of the origin.
- The mathematical expressions for the sifting property is shown below .Sifting property is the property that sifts out the signal.
[Tex]\int_{-\infty }^{\infty } y(t).\delta (t-to) dt = y (to)[/Tex] in continuous time domain where to is a constant value .
[Tex]\sum_{n=-\infty }^{\infty } y[n] \delta [n-K] = y[K][/Tex] in discrete time domain where K is a constant.
- Unit impulse exhibits the convolution identity property . Convolution is combining of two or more signals to get a third signal . The mathematical expressions for the convolution property of unit impulse signal in both continuous time domain and discrete time domain are represented below:
[Tex]y(t)*\delta (t) = y(t)[/Tex] in continous time domain representation .
[Tex]y[n]*\delta [n] = y[n][/Tex] in discrete time domain representation .
- Unit impulse signal has the property of linearity in both continuous time and discrete time domains. Linearity is the property of the system that follows the prniciple of superposition.
Unit Impulse Signal in Control System
In this article, we are going to learn about the unit impulse signal in control systems. We know that a signal is a source of information that varies with time, space, temperature, and other independent variables. We can also define a signal as a function that conveys information about a phenomenon or a process. There are two types of signals continuous time domain signals that are defined for every instant of time and discrete time domain signals that are defined for discrete time instants. Standard signals are the signals that are used to know the performance of the control systems using the time response of the output. These signals have the known form of the mathematical expression. Standard signals are impulse, step, ramp, parabolic, and sinusoidal. Unit impulse signal is also one of the standard signals. Let us now discuss about the impulse and unit impulse signal in detail.
Table of Content
- What is a Unit Impulse Signal?
- Types of unit impulse signal
- Properties of unit impulse signal
- Characteristics of unit impulse signal
- Applications of unit impulse signal
- Advantages of unit impulse signal
- Disadvantages of unit impulse signal